Diffusion of Neon in White Dwarf Stars J. Hughto,∗ A. S. Schneider, and C. J. Horowitz† Department of Physics and Nuclear Theory Center, Indiana University, Bloomington, IN 47405

D. K. Berry

arXiv:1009.4248v2 [astro-ph.SR] 29 Nov 2010

University Information Technology Services, Indiana University, Bloomington, IN 47408 (Dated: November 30, 2010) Sedimentation of the neutron rich isotope 22 Ne may be an important source of gravitational energy during the cooling of white dwarf stars. This depends on the diffusion constant for 22 Ne in strongly coupled plasma mixtures. We calculate self-diffusion constants Di from molecular dynamics simulations of carbon, oxygen, and neon mixtures. We find that Di in a mixture does not differ greatly from earlier one component plasma results. For strong coupling (coulomb parameter Γ > −2/3 few), Di has a modest dependence on the charge Zi of the ion species, Di ∝ Zi . However Di depends more strongly on Zi for weak coupling (smaller Γ). We conclude that the self-diffusion constant DNe for 22 Ne in carbon, oxygen, and neon plasma mixtures is accurately known so that uncertainties in DNe should be unimportant for simulations of white dwarf cooling. PACS numbers: 94.05.-a, 97.20.Rp, 97.60.-s, 66.10.C-

I.

INTRODUCTION

Observations of cooling white dwarf (WD) stars provide important ways to date stellar systems [1]. White dwarf cooling involves simple physics because the energy from nuclear reactions is modest. However, chemical energy from the latent heat of fusion and gravitational energy from sedimentation can be important. Understanding these additional energy sources may allow more accurate stellar dates, see for example [2], and provide additional information on the interior composition of WDs [3]. The interior of a WD is a coulomb plasma of ions and a degenerate electron gas. As the star cools this plasma crystallizes. Winget et al. recently observed effects from the latent heat of crystallization on the luminosity function of WDs in the globular cluster NGC 6397 [4]. Winget et al.’s observations constrain the melting temperature of the carbon and oxygen mixtures expected in these WD cores. This temperature depends on the ratio of carbon to oxygen. Therefore observations of crystallization may provide information on WD composition. The ratio of carbon to oxygen in WD stars is very interesting. It depends on the rate for the reaction 12 C(α, γ)16 O. Despite a great deal of effort, see for example [5], the stellar rate for this reaction remains one of the most important unsettled rates left in Nuclear Astrophysics [6]. Furthermore, the ratio of carbon to oxygen in massive stars is important for their subsequent evolution and nucleosynthesis [7]. Therefore, a measurement of the carbon to oxygen ratio in a WD could be very important. To determine the C/O ratio from observations of the melting temperature, one needs the phase diagram for

∗ Electronic † Electronic

address: [email protected] address: [email protected]

carbon and oxygen mixtures. Recently, we determined this phase diagram using molecular dynamics simulations and concluded that, if the melting temperature is close to that for pure carbon, then the central oxygen fraction by mass of the carbon/oxygen WDs in NGC 6397 is less than about 64 % [3]. Note that the interior of WDs can also be probed with astro-seismology, see for example ref. [8]. Sedimentation of 22 Ne could provide an additional energy source during WD cooling [9, 10]. Much of the carbon, nitrogen, and oxygen, originally present in the star, is converted by nuclear reactions into 22 Ne. This neutron rich isotope, with 12 neutrons and 10 protons, has a larger mass to charge ratio than 12 C or 16 O. As a result it will sink in the strong gravitational field of the star and release gravitational energy. Recently, Garcia-Berro et al. [11] studied the effects of 22 Ne on WD evolution. Sedimentation could be most important in very metal rich stars that have more 22 Ne. Both the release of latent heat from crystallization and gravitational energy from sedimentation can delay WD cooling. Garcia-Berro et al. [2] can explain the long 8 Gyr age for the metal rich open star cluster NGC 6791 by including both crystallization and sedimentation. Furthermore, it may be possible to separate these two effects by comparing observations of metal rich systems such as NGC 6791 to less metal rich systems such as NGC 6397 where sedimentation should be unimportant. Sedimentation depends on the diffusion constant D for 22 Ne ions in strongly coupled plasma mixtures. There have been previous calculations of D, starting with the MD simulations of Hansen et al. for the one component plasma (OCP) [12]. The one component plasma consists of ions, with pure coulomb interactions, and an inert neutralizing background charge density. Diffusion in the OCP in a strong magnetic field was considered by Bernu [13]. Hansen et al. have also calculated diffusion for binary mixtures [14].

2 Diffusion for a Yukawa fluid has been simulated by Robbins et al. [15] and Ohta et al. [16]. In a Yukawa fluid ions interact via a screened coulomb potential vij (r), vij (r) =

Zi Zj e2 −r/λ e , r

(1)

for two ions with charges Zi and Zj , that are separated by a distance r. The OCP is equivalent to a Yukawa fluid, where all of the ions have the same charge and the screening length λ is very large. The motion of carbon, oxygen, and neon ions in a WD is largely classical because of their large masses. However, at great densities there could be some quantum corrections that might increase D. These have been estimated by Daligault and Murillo [17], in a semiclassical model, and found to be very small. Instead, these authors argue that accurate classical simulations of diffusion coefficients in strongly coupled mixtures with impurities would be useful for WD cooling. These diffusion coefficients would also be useful to describe sedimentation in neutron stars, see for example [18]. In this paper, we present classical MD simulations of carbon, oxygen, and neon plasma mixtures in order to determine diffusion coefficients D. Our results are more accurate than previous work because we explicitly simulate mixtures with realistic WD compositions. In Section II we describe our MD formalism and present results for diffusion coefficents in Section III. We conclude in Section IV. II.

FORMALISM

We describe our MD simulation formalism. This is similar to what we used earlier to simulate the carbon/ oxygen phase diagram in ref. [3]. We consider a three component mixture of carbon (12 C), oxygen (16 O), and neon (22 Ne), were the neon is assumed to have a small concentration. A star with near solar metallicity, that has most of its original carbon, nitrogen, and oxygen converted into 22 Ne, might have of order 2% 22 Ne. The ratio of carbon to oxygen in the WD core depends on the rates for the 4 He(2α, γ)12 C and 12 C(α, γ)16 O reactions and is expected to be near one to one, see for example [3, 20]. Therefore in this paper we present simulations using a mixture of 49% 12 C, 49% 16 O, and 2% 22 Ne, by number. We expect our results for the Ne self-diffusion coefficient to be nearly independent of 22 Ne concentration, as long as it is small. Likewise we do not expect much dependence of DNe on the ratio of carbon to oxygen. In addition to this three component mixture, we also present results for a one component system of pure oxygen for comparison. The ions are assumed to interact via screened Yukawa interactions, see Eq. 1. The Thomas Fermi screening length λ, for cold relativistic electrons, is λ−1 = 2α1/2 kF /π 1/2 where the electron Fermi momentum kF is kF = (3π 2 ne )1/3 and α is the fine structure constant.

The electron density ne is equal to the ion charge density, ne = hZin, where n is the ion density and hZi is the average charge. Our simulations are classical and we have neglected the electron mass (extreme relativistic limit). This is to be consistent with our previous work on neutron stars. However, the electron mass is important at the lower densities in WD and this may change our results slightly [21]. Also quantum effects could play some role at high densities [22],[23]. For relativistic electrons, the ratio of λ to the ion sphere radius a, a=

3 1/3 , 4πn

(2)

depends only on the average charge hZi. For our three component mixture we use λ = 2.816a, while for pure oxygen, we use λ = 2.703a. For nonrelativistic electrons λ/a can be somewhat smaller. In Sec. III we find that our results for diffusion constants D are insensitive to a fifty percent decrease in λ. The simulations can be characterized by an average coulomb parameter Γ, Γ=

hZ 5/3 ie2 . ae T

(3)

Here hZ 5/3 i is an average over the ion charges, T is the temperature, and the electron sphere radius ae is ae = (3/4πne )1/3 with ne = hZin the electron density. The pure system freezes near Γ = 175 [21] while the C/O/Ne mixture is expected to freeze at a somewhat higher Γ [3, 24]. Note that these values of Γ may depend slightly on λ [21, 25]. Time can be measured in our simulations in units of one over the plasma frequency ωp . Long wavelength fluctuations in the charge density can undergo oscillations at the plasma frequency. This depends on the ion charge Z and mass M . For mixtures we define a hydrodynamical plasma frequency ω ¯ p from the simple averages of Z and M, ω ¯p =

h 4πe2 hZi2 n i1/2 hM i

.

(4)

Note that other choices for the average over composition in Eq. 4 are possible. However, they are expected to give very similar results for the average plasma frequency. Self diffusion constants Di , for ions of type i, are calculated from the velocity autocorrelation function Z i (t), Z i (t) =

hvj (t0 + t) · vj (t0 )i hvj (t0 ) · vj (t0 )i

(5)

where the average is over all ions j of a given type and over initial times t0 . The velocity of the jth ion at time t is vj (t). The diffusion constant is calculated from the time integral of Z i (t), Z tmax T Di = dtZ i (t). (6) Mi 0

3

III.

RESULTS

We now present results for the velocity autocorrelation function Z i (t) and diffusion constant Di , first for a mixture of carbon, oxygen, and neon and then for a pure oxygen system. Fig. 1 shows the velocity autocorrelation function for 22 Ne from a simulation with N = 8192 ions. The velocity autocorrelation function oscillates with frequency near ω ¯ p and the amplitude of these oscillations increases with Γ. Note that at Γ = 244 the system is still in a (possibly metastable) fluid state. In Fig. 2 we show Z i (t) for i =12 C, 16 O, and 22 Ne ions. The difference in Z i for different ions is subtle. We see that Z i (t) at t near 5/¯ ωp is more negative for 12 C 22 than for Ne. Perhaps the lighter carbon ions bounce backwards from the confining cage of other ions with a more negative velocity than do neon ions. These subtle differences in Fig. 2, along with the explicit factor of 1/Mi in Eq. 6, lead to differences in diffusion constants for different ions. We choose to scale our diffusion results with Hansen et al’s simple fit to their original MD results for the diffusion constant D0 of a one component plasma [12]. D0 =

3¯ ωp a2 Γ4/3

(7)

Note that we have generalized ref. [12] results to a multicomponent system by using the average plasma frequency ω ¯ p in Eq. 7. In Table I and Fig. 3 we present our MD simulation results for Di /D0 . We see that Eq. 7 is indeed a reasonable first approximation to Di and our results only differ from this by about 50% or less. Finite size effects appear to be modest with 3456 ion simulation

1 0.8

Γ=9.8 39 98 244

0.6 0.4

Ne

Z (t)

We find that Z i (t) is small by a maximum time of tmax = 220/¯ ωp . We start our MD simulations from a random configuration at a relatively high temperature so that Γ ≈ 1. We evolve the system in time using the simple velocity Verlet algorithm [26] with a time step ∆t = 1/18¯ ωp . We use periodic boundary conditions. We do not use a cutoff in the interaction at large distances and evaluate the force on a given ion by summing over all of the other ions in the simulation. We test for finite size effects by performing simulations for N = 3456, 8192, and 27648 ions. Our largest simulation volume is large enough so that one half of the box length L is much larger than the electron screening length λ, L/2 = 8.65λ. We approximately maintain the system at a given temperature by simply rescaling the velocities every 10 time steps. Typically we equilibrate the system by evolving for a time 2200/¯ ωp . Then we take data for an additional time of 2200/¯ ωp by writing the velocities to disk every 1/9¯ ωp (two time steps). Next we rescale the velocities to a lower temperature and repeat the equilibration and data taking steps for the next higher value of Γ. We present results for Z i (t) and Di in Section III

0.2 0 -0.2 -0.4 0

10

20

t ωp

30

40

50

FIG. 1: (Color on line) Velocity autocorrelation function Z i (t) versus time t for 22 Ne ions in a carbon, oxygen, and neon mixture. The N=8192 ion system is in a fluid state and the curves are for values of the coulomb parameter Γ =9.8 (solid), 39 (dotted), 98 (dashed), and 244 (dot-dashed).

results being below 8192 or 27648 ion results by only a few %. In general there is good agreement between 8192 and 27648 ion results. However, our results for 22 Ne have larger statistical errors than for 12 C or 16 O because there are fewer 22 Ne ions in our simulations, since its abundance is assumed to be only 2%. We test the sensitivity of our results to the screening length λ by running a 27648 ion C, O, Ne system with a fifty percent smaller λ. We find at Γ = 9.75 that diffusion constants Di only increase by two percent or less. The dependence of Di on ion species is very interesting. The 12 C, 16 O, and 22 Ne results in Fig. 3 are nearly parallel, for Γ > few, and depend on the ion charge Zi −2/3 approximately as Zi . In general, Di depends on both the mass and the charge of the ion. However, since the ions in our simulations have similar charge to mass ratios we do not separate out these two effects and the observed −2/3 Zi dependance includes both effects. We can approximately fit all of the results in Fig. 3 with Γ > 5 with a simple expression, h hZi i 23 Di ≈ 0.53 (1 + 0.22Γ) exp(−0.135Γ0.62 ). D0 Zi

(8)

This Eq. is shown as dotted lines in Fig. 3. For weak coupling (small Γ) the dependance of Di on Zi is stronger. In the limit of very weak coupling, Di may be related to a mean free path which depends on one over an interaction cross section. In Born approximation this cross section scales as Zi2 and gives a Di that depends −2/3 more strongly on Zi than Zi . We illustrate this by plotting the same data of Fig. 3 but as a function of 1/Γ instead of Γ in order to show the low Γ data more clearly.

4 TABLE I: Diffusion constants Di /D0 , for i=12 C, 16 O, and Ne from simulations with N = 3456, 8192 and 27648 ions, for different coulomb parameter Γ values. The diffusion constants are scaled with D0 = 3¯ ωp a2 /Γ4/3 which is a simple fit to the original one component plasma results, see Eq. 7 in the text. 12 Γ N C 16 O 22 Ne 22

1.392

1.948

2.783

4.871

9.742

19.48

38.97

64.95

97.42

147.6

194.84

221.40

243.55

Ne O C

0.6 0.4

i

Z (t)

3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648

0.8

0.2

1.092 0.717 0.460

0

0.852 0.827 0.827 0.832

0.833 0.894 0.881 0.907 1.081 1.058 1.087 1.311 1.343 1.327 1.455 1.464 1.488 1.446 1.420 1.446 1.294 1.297 1.295

0.582 0.580 0.587 0.586

0.611 0.697 0.703 0.706 0.888 0.895 0.904 1.121 1.126 1.137 1.251 1.232 1.278 1.216 1.229 1.228 1.077 1.082 1.093

0.449 0.477 0.460 0.453

0.500 0.572 0.625 0.609 0.749 0.758 0.788 0.995 1.024 0.984 1.114 1.098 1.119 1.076 1.120 1.100 0.940 0.918 0.979

1.034 0.859 0.718 0.779 0.628 0.547 0.783 0.638 0.533

-0.2 0

5

10

t ωp

15

20

25

FIG. 2: (Color on line) Velocity autocorrelation function Z i (t) versus time t for i =22 Ne (solid cuve), 16 O (dotted), and 12 C (dashed) ions. The simulation has N = 8192 ions at coulomb parameter Γ = 195.2 and is in a liquid state.

1.5

Di/D0

0.974

1

N=27648 N=8192 Fit

1

0.5 0

50

100

Γ

150

200

250

FIG. 3: (Color on line) Diffusion constants Di over a simple fit to the original MD results for the one component plasma D0 , see Eq. 7, versus coulomb parameter Γ. Results for 12 C (solid black line), 16 O (dashed red), and 22 Ne (dot-dashed blue) are presented for a simulation with 27648 ions. Results for 8192 ions (squares) are also shown. Finally the dotted lines show the simple global fit of Eq. 8.

0.646 0.516 0.443 0.526 0.408 0.337 0.544 0.433 0.352

We see that Di /D0 increases for 12 C for small Γ in a way that is not described by Eq. 8. Indeed Fig. 4 shows that Di depends more strongly on Zi than Eq. 8 for small Γ. Finally we also perform MD simulations for a one component system of pure 16 O in order to compare to our multicomponent results. Results for MD simulations with 8192 and 27648 ions are shown in Table II

5

1.4 N=27648 N=8192 Fit

N=8192 27648 Fit

1.2 1 D/D0

Di/D0

1.5

1

0.8 0.6

0.5 0

0.4 0

1

0.5 −1 Γ

FIG. 4: (Color on line) Diffusion constants Di over D0 , see Eq. 7, versus the inverse of the coulomb parameter 1/Γ. Results for 12 C (solid black line), 16 O (dashed red), and 22 Ne (dotdashed blue) are presented for a simulation with 27648 ions. Results for 8192 ions (squares) are also shown. Finally the dotted lines show the simple global fit of Eq. 8. This disagrees with 12 C and 16 O data for small Γ

50

100

Γ

150

200

250

FIG. 5: (Color on line) Diffusion constant Di over D0 , see Eq. 7, versus the coulomb parameter Γ for pure oxygen. Results for a simulation with 27648 ions are shown as circles connected by a solid black line while the red squares show 8192 ion results. Finally the dashed line shows the simple global fit of Eq. 8 with hZi/Zi = 1.

IV.

CONCLUSIONS

16

TABLE II: Diffusion constant D16 O /D0 , for a pure O system from simulations with N = 8192 and 27648 ions in a (possibly metastable) fluid state. The diffusion constant is scaled with D0 = 3¯ ωp a2 /Γ4/3 . This is a simple fit to the original one component plasma results, see Eq. 7 in the text. Γ 2.470 6.175 12.35 24.70 49.40 82.33 123.50 187.12 247.00

D/D0 D/D0 N = 8192 N=27648 0.680 0.831 1.035 1.237 1.307 1.194 1.000 .439

0.828 1.045 1.274 1.330 1.223 1.014 0.703 0.441

and Fig. 5. There is good agreement between these two simulations and reasonable agreement with Eq. 8 with hZi/Zi = 1. Previous MD simulations for one component Yukawa fluids fit D with [15, 17, 19] 173 1.154 D = 0.0028 + 0.00525 − 1 . ωp a2 Γ

(9)

The ratio of Eq. 9 to Eq. 7 is very similar in form to Fig. 5, see Fig. 5 of Ref. [17]. However, we find a somewhat larger amplitude for the deviations of D from D0 .

Sedimentation of the neutron rich isotope 22 Ne may be an important source of gravitational energy during the cooling of white dwarf stars. This depends on the diffusion constant for 22 Ne in strongly coupled plasma mixtures. We have calculated self-diffusion constants Di from molecular dynamics simulations of carbon, oxygen, and neon mixtures. We find that, for strong coupling (coulomb parameter Γ > few), Di has a modest dependence on the charge −2/3 Zi of the ion species, Di ∝ Zi . However Di depends more strongly on Zi for weak coupling (smaller Γ). Our results for both a carbon, oxygen, neon mixture, and for a one component plasma can be fit by, 1 + 0.22Γ hZi 2/3 Di 0.62 ≈ 1.59 exp(−0.135Γ ) , ω ¯ p a2 Zi Γ4/3 (10) for Γ > few, see Eqs 8,7. Here ω ¯ p is the average plasma frequency and a is the ion sphere radius. We conclude that the self-diffusion constant DN e for 22 Ne in carbon, oxygen plasma mixtures is accurately known so that uncertainties in DN e should be unimportant for simulations of white dwarf cooling. All of our diffusion results have been for the liquid phase. We are not aware of any results for Di in solids. Often Di is arbitrarily assumed to be zero for a solid. In future work we will simulate Di in the solid phase, both for single component and multicomponent systems. We thank E. Brown, A. Cumming, and Z. Medin for helpful discussions. This research was supported in part

6 by DOE grant DE-FG02-87ER40365 and by the National Science Foundation through TeraGrid resources provided by National Institute for Computational Sciences, and

Texas Advanced Computing Center under grant TGAST100014.

[1] G. Fontaine, P. Brassard, P. Bergeron, Proc. Astronomical Society Pacific 113, 409 (2001). [2] E. Garc´ıa-Berro et al., Nature 465, 194 (2010). [3] C. J. Horowitz, A. S. Schneider, D. K. Berry, Phys Rev Lett, 104, 231101 (2010). [4] D. E. Winget et al., Astrophys. J. 693, L6 (2009). [5] R. Kunz et al., Astrophys. J. 567, 643 (2002). [6] L. R. Buchmann, C. A. Barnes, Nuc. Phys. A, 777, 254 (2006). [7] M. F. El Eid, B. S. Meyer, L. S. The, Astrophys. J. 611, 452 (2004). [8] T. S. Metcalfe, Monthly Notice Royal Astron. Soc. 363, L86 (2005). [9] Lars Bildsten, David M. Hall, Astrophys. J. 549, L219 (2001). [10] C. J. Deloye, L. Bildsten, Astrophys. J. 580, 1077 (2002). [11] E. Garcia-Berro, L. G. Althaus, A. H. Corsico, J. Isern, Astrophysical J. 677, 473 (2008). [12] J. P. Hansen, I. R. McDonald, E. L. Pollock, Phys. Rev. A11, 1025 (1975). [13] B. Bernu, J. Physique Letters, 42, 253 (1981). [14] J. R. Hansen, F. Joly, I. R. McDonald, Physica 132A, 472 (1985). [15] M. O. Robbins, K. Krener, G. S. Grest, J. Chem. Phys.

88, 3286 (1988). [16] H. Ohta, S. Hamaguchi, Physics of Plasmas, 7, 4506 (2000). [17] J. Daligault, M. S. Murillo, Phys. Rev. E71, 036408 (2005). [18] Fang Peng, Edward F. Brown, James W. Truran, Astrophys. J. 654, 1022 (2007). [19] S. Ranganathan, R. E. Johnson, and C. E. Woodward, Phys. Chem. Liq. 41, 123 (2003). [20] M. Salaris et al., Astrophys. J. 486, 413 (1997). [21] A. Y. Potekhin, G. Chabrier, Phys. Rev. E 62, 8554 (2000). [22] A. Y. Potekhin, Doctor of Science thesis (in Russian) http://www.ioffe.ru/astro/DTA/palex/disser.pdf and A. Y. Potekhin, G. Chabrier, Contrib. Plasma Physics 50, 82 (2010). [23] M. D. Jones, D. M. Ceperley, Phys. Rev. Lett. 76, 4572 (1996). [24] L. Segretain, Astronomy & Astrophys. 310, 485 (1996). [25] S. Hamaguchi, R. T. Farouki, D. H. E. Dubin, Phys. Rev. E 56, 4671 (1997). [26] L. Verlet, Phys. Rev. 159, 98 (1967).

D. K. Berry

arXiv:1009.4248v2 [astro-ph.SR] 29 Nov 2010

University Information Technology Services, Indiana University, Bloomington, IN 47408 (Dated: November 30, 2010) Sedimentation of the neutron rich isotope 22 Ne may be an important source of gravitational energy during the cooling of white dwarf stars. This depends on the diffusion constant for 22 Ne in strongly coupled plasma mixtures. We calculate self-diffusion constants Di from molecular dynamics simulations of carbon, oxygen, and neon mixtures. We find that Di in a mixture does not differ greatly from earlier one component plasma results. For strong coupling (coulomb parameter Γ > −2/3 few), Di has a modest dependence on the charge Zi of the ion species, Di ∝ Zi . However Di depends more strongly on Zi for weak coupling (smaller Γ). We conclude that the self-diffusion constant DNe for 22 Ne in carbon, oxygen, and neon plasma mixtures is accurately known so that uncertainties in DNe should be unimportant for simulations of white dwarf cooling. PACS numbers: 94.05.-a, 97.20.Rp, 97.60.-s, 66.10.C-

I.

INTRODUCTION

Observations of cooling white dwarf (WD) stars provide important ways to date stellar systems [1]. White dwarf cooling involves simple physics because the energy from nuclear reactions is modest. However, chemical energy from the latent heat of fusion and gravitational energy from sedimentation can be important. Understanding these additional energy sources may allow more accurate stellar dates, see for example [2], and provide additional information on the interior composition of WDs [3]. The interior of a WD is a coulomb plasma of ions and a degenerate electron gas. As the star cools this plasma crystallizes. Winget et al. recently observed effects from the latent heat of crystallization on the luminosity function of WDs in the globular cluster NGC 6397 [4]. Winget et al.’s observations constrain the melting temperature of the carbon and oxygen mixtures expected in these WD cores. This temperature depends on the ratio of carbon to oxygen. Therefore observations of crystallization may provide information on WD composition. The ratio of carbon to oxygen in WD stars is very interesting. It depends on the rate for the reaction 12 C(α, γ)16 O. Despite a great deal of effort, see for example [5], the stellar rate for this reaction remains one of the most important unsettled rates left in Nuclear Astrophysics [6]. Furthermore, the ratio of carbon to oxygen in massive stars is important for their subsequent evolution and nucleosynthesis [7]. Therefore, a measurement of the carbon to oxygen ratio in a WD could be very important. To determine the C/O ratio from observations of the melting temperature, one needs the phase diagram for

∗ Electronic † Electronic

address: [email protected] address: [email protected]

carbon and oxygen mixtures. Recently, we determined this phase diagram using molecular dynamics simulations and concluded that, if the melting temperature is close to that for pure carbon, then the central oxygen fraction by mass of the carbon/oxygen WDs in NGC 6397 is less than about 64 % [3]. Note that the interior of WDs can also be probed with astro-seismology, see for example ref. [8]. Sedimentation of 22 Ne could provide an additional energy source during WD cooling [9, 10]. Much of the carbon, nitrogen, and oxygen, originally present in the star, is converted by nuclear reactions into 22 Ne. This neutron rich isotope, with 12 neutrons and 10 protons, has a larger mass to charge ratio than 12 C or 16 O. As a result it will sink in the strong gravitational field of the star and release gravitational energy. Recently, Garcia-Berro et al. [11] studied the effects of 22 Ne on WD evolution. Sedimentation could be most important in very metal rich stars that have more 22 Ne. Both the release of latent heat from crystallization and gravitational energy from sedimentation can delay WD cooling. Garcia-Berro et al. [2] can explain the long 8 Gyr age for the metal rich open star cluster NGC 6791 by including both crystallization and sedimentation. Furthermore, it may be possible to separate these two effects by comparing observations of metal rich systems such as NGC 6791 to less metal rich systems such as NGC 6397 where sedimentation should be unimportant. Sedimentation depends on the diffusion constant D for 22 Ne ions in strongly coupled plasma mixtures. There have been previous calculations of D, starting with the MD simulations of Hansen et al. for the one component plasma (OCP) [12]. The one component plasma consists of ions, with pure coulomb interactions, and an inert neutralizing background charge density. Diffusion in the OCP in a strong magnetic field was considered by Bernu [13]. Hansen et al. have also calculated diffusion for binary mixtures [14].

2 Diffusion for a Yukawa fluid has been simulated by Robbins et al. [15] and Ohta et al. [16]. In a Yukawa fluid ions interact via a screened coulomb potential vij (r), vij (r) =

Zi Zj e2 −r/λ e , r

(1)

for two ions with charges Zi and Zj , that are separated by a distance r. The OCP is equivalent to a Yukawa fluid, where all of the ions have the same charge and the screening length λ is very large. The motion of carbon, oxygen, and neon ions in a WD is largely classical because of their large masses. However, at great densities there could be some quantum corrections that might increase D. These have been estimated by Daligault and Murillo [17], in a semiclassical model, and found to be very small. Instead, these authors argue that accurate classical simulations of diffusion coefficients in strongly coupled mixtures with impurities would be useful for WD cooling. These diffusion coefficients would also be useful to describe sedimentation in neutron stars, see for example [18]. In this paper, we present classical MD simulations of carbon, oxygen, and neon plasma mixtures in order to determine diffusion coefficients D. Our results are more accurate than previous work because we explicitly simulate mixtures with realistic WD compositions. In Section II we describe our MD formalism and present results for diffusion coefficents in Section III. We conclude in Section IV. II.

FORMALISM

We describe our MD simulation formalism. This is similar to what we used earlier to simulate the carbon/ oxygen phase diagram in ref. [3]. We consider a three component mixture of carbon (12 C), oxygen (16 O), and neon (22 Ne), were the neon is assumed to have a small concentration. A star with near solar metallicity, that has most of its original carbon, nitrogen, and oxygen converted into 22 Ne, might have of order 2% 22 Ne. The ratio of carbon to oxygen in the WD core depends on the rates for the 4 He(2α, γ)12 C and 12 C(α, γ)16 O reactions and is expected to be near one to one, see for example [3, 20]. Therefore in this paper we present simulations using a mixture of 49% 12 C, 49% 16 O, and 2% 22 Ne, by number. We expect our results for the Ne self-diffusion coefficient to be nearly independent of 22 Ne concentration, as long as it is small. Likewise we do not expect much dependence of DNe on the ratio of carbon to oxygen. In addition to this three component mixture, we also present results for a one component system of pure oxygen for comparison. The ions are assumed to interact via screened Yukawa interactions, see Eq. 1. The Thomas Fermi screening length λ, for cold relativistic electrons, is λ−1 = 2α1/2 kF /π 1/2 where the electron Fermi momentum kF is kF = (3π 2 ne )1/3 and α is the fine structure constant.

The electron density ne is equal to the ion charge density, ne = hZin, where n is the ion density and hZi is the average charge. Our simulations are classical and we have neglected the electron mass (extreme relativistic limit). This is to be consistent with our previous work on neutron stars. However, the electron mass is important at the lower densities in WD and this may change our results slightly [21]. Also quantum effects could play some role at high densities [22],[23]. For relativistic electrons, the ratio of λ to the ion sphere radius a, a=

3 1/3 , 4πn

(2)

depends only on the average charge hZi. For our three component mixture we use λ = 2.816a, while for pure oxygen, we use λ = 2.703a. For nonrelativistic electrons λ/a can be somewhat smaller. In Sec. III we find that our results for diffusion constants D are insensitive to a fifty percent decrease in λ. The simulations can be characterized by an average coulomb parameter Γ, Γ=

hZ 5/3 ie2 . ae T

(3)

Here hZ 5/3 i is an average over the ion charges, T is the temperature, and the electron sphere radius ae is ae = (3/4πne )1/3 with ne = hZin the electron density. The pure system freezes near Γ = 175 [21] while the C/O/Ne mixture is expected to freeze at a somewhat higher Γ [3, 24]. Note that these values of Γ may depend slightly on λ [21, 25]. Time can be measured in our simulations in units of one over the plasma frequency ωp . Long wavelength fluctuations in the charge density can undergo oscillations at the plasma frequency. This depends on the ion charge Z and mass M . For mixtures we define a hydrodynamical plasma frequency ω ¯ p from the simple averages of Z and M, ω ¯p =

h 4πe2 hZi2 n i1/2 hM i

.

(4)

Note that other choices for the average over composition in Eq. 4 are possible. However, they are expected to give very similar results for the average plasma frequency. Self diffusion constants Di , for ions of type i, are calculated from the velocity autocorrelation function Z i (t), Z i (t) =

hvj (t0 + t) · vj (t0 )i hvj (t0 ) · vj (t0 )i

(5)

where the average is over all ions j of a given type and over initial times t0 . The velocity of the jth ion at time t is vj (t). The diffusion constant is calculated from the time integral of Z i (t), Z tmax T Di = dtZ i (t). (6) Mi 0

3

III.

RESULTS

We now present results for the velocity autocorrelation function Z i (t) and diffusion constant Di , first for a mixture of carbon, oxygen, and neon and then for a pure oxygen system. Fig. 1 shows the velocity autocorrelation function for 22 Ne from a simulation with N = 8192 ions. The velocity autocorrelation function oscillates with frequency near ω ¯ p and the amplitude of these oscillations increases with Γ. Note that at Γ = 244 the system is still in a (possibly metastable) fluid state. In Fig. 2 we show Z i (t) for i =12 C, 16 O, and 22 Ne ions. The difference in Z i for different ions is subtle. We see that Z i (t) at t near 5/¯ ωp is more negative for 12 C 22 than for Ne. Perhaps the lighter carbon ions bounce backwards from the confining cage of other ions with a more negative velocity than do neon ions. These subtle differences in Fig. 2, along with the explicit factor of 1/Mi in Eq. 6, lead to differences in diffusion constants for different ions. We choose to scale our diffusion results with Hansen et al’s simple fit to their original MD results for the diffusion constant D0 of a one component plasma [12]. D0 =

3¯ ωp a2 Γ4/3

(7)

Note that we have generalized ref. [12] results to a multicomponent system by using the average plasma frequency ω ¯ p in Eq. 7. In Table I and Fig. 3 we present our MD simulation results for Di /D0 . We see that Eq. 7 is indeed a reasonable first approximation to Di and our results only differ from this by about 50% or less. Finite size effects appear to be modest with 3456 ion simulation

1 0.8

Γ=9.8 39 98 244

0.6 0.4

Ne

Z (t)

We find that Z i (t) is small by a maximum time of tmax = 220/¯ ωp . We start our MD simulations from a random configuration at a relatively high temperature so that Γ ≈ 1. We evolve the system in time using the simple velocity Verlet algorithm [26] with a time step ∆t = 1/18¯ ωp . We use periodic boundary conditions. We do not use a cutoff in the interaction at large distances and evaluate the force on a given ion by summing over all of the other ions in the simulation. We test for finite size effects by performing simulations for N = 3456, 8192, and 27648 ions. Our largest simulation volume is large enough so that one half of the box length L is much larger than the electron screening length λ, L/2 = 8.65λ. We approximately maintain the system at a given temperature by simply rescaling the velocities every 10 time steps. Typically we equilibrate the system by evolving for a time 2200/¯ ωp . Then we take data for an additional time of 2200/¯ ωp by writing the velocities to disk every 1/9¯ ωp (two time steps). Next we rescale the velocities to a lower temperature and repeat the equilibration and data taking steps for the next higher value of Γ. We present results for Z i (t) and Di in Section III

0.2 0 -0.2 -0.4 0

10

20

t ωp

30

40

50

FIG. 1: (Color on line) Velocity autocorrelation function Z i (t) versus time t for 22 Ne ions in a carbon, oxygen, and neon mixture. The N=8192 ion system is in a fluid state and the curves are for values of the coulomb parameter Γ =9.8 (solid), 39 (dotted), 98 (dashed), and 244 (dot-dashed).

results being below 8192 or 27648 ion results by only a few %. In general there is good agreement between 8192 and 27648 ion results. However, our results for 22 Ne have larger statistical errors than for 12 C or 16 O because there are fewer 22 Ne ions in our simulations, since its abundance is assumed to be only 2%. We test the sensitivity of our results to the screening length λ by running a 27648 ion C, O, Ne system with a fifty percent smaller λ. We find at Γ = 9.75 that diffusion constants Di only increase by two percent or less. The dependence of Di on ion species is very interesting. The 12 C, 16 O, and 22 Ne results in Fig. 3 are nearly parallel, for Γ > few, and depend on the ion charge Zi −2/3 approximately as Zi . In general, Di depends on both the mass and the charge of the ion. However, since the ions in our simulations have similar charge to mass ratios we do not separate out these two effects and the observed −2/3 Zi dependance includes both effects. We can approximately fit all of the results in Fig. 3 with Γ > 5 with a simple expression, h hZi i 23 Di ≈ 0.53 (1 + 0.22Γ) exp(−0.135Γ0.62 ). D0 Zi

(8)

This Eq. is shown as dotted lines in Fig. 3. For weak coupling (small Γ) the dependance of Di on Zi is stronger. In the limit of very weak coupling, Di may be related to a mean free path which depends on one over an interaction cross section. In Born approximation this cross section scales as Zi2 and gives a Di that depends −2/3 more strongly on Zi than Zi . We illustrate this by plotting the same data of Fig. 3 but as a function of 1/Γ instead of Γ in order to show the low Γ data more clearly.

4 TABLE I: Diffusion constants Di /D0 , for i=12 C, 16 O, and Ne from simulations with N = 3456, 8192 and 27648 ions, for different coulomb parameter Γ values. The diffusion constants are scaled with D0 = 3¯ ωp a2 /Γ4/3 which is a simple fit to the original one component plasma results, see Eq. 7 in the text. 12 Γ N C 16 O 22 Ne 22

1.392

1.948

2.783

4.871

9.742

19.48

38.97

64.95

97.42

147.6

194.84

221.40

243.55

Ne O C

0.6 0.4

i

Z (t)

3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648 3456 8192 27648

0.8

0.2

1.092 0.717 0.460

0

0.852 0.827 0.827 0.832

0.833 0.894 0.881 0.907 1.081 1.058 1.087 1.311 1.343 1.327 1.455 1.464 1.488 1.446 1.420 1.446 1.294 1.297 1.295

0.582 0.580 0.587 0.586

0.611 0.697 0.703 0.706 0.888 0.895 0.904 1.121 1.126 1.137 1.251 1.232 1.278 1.216 1.229 1.228 1.077 1.082 1.093

0.449 0.477 0.460 0.453

0.500 0.572 0.625 0.609 0.749 0.758 0.788 0.995 1.024 0.984 1.114 1.098 1.119 1.076 1.120 1.100 0.940 0.918 0.979

1.034 0.859 0.718 0.779 0.628 0.547 0.783 0.638 0.533

-0.2 0

5

10

t ωp

15

20

25

FIG. 2: (Color on line) Velocity autocorrelation function Z i (t) versus time t for i =22 Ne (solid cuve), 16 O (dotted), and 12 C (dashed) ions. The simulation has N = 8192 ions at coulomb parameter Γ = 195.2 and is in a liquid state.

1.5

Di/D0

0.974

1

N=27648 N=8192 Fit

1

0.5 0

50

100

Γ

150

200

250

FIG. 3: (Color on line) Diffusion constants Di over a simple fit to the original MD results for the one component plasma D0 , see Eq. 7, versus coulomb parameter Γ. Results for 12 C (solid black line), 16 O (dashed red), and 22 Ne (dot-dashed blue) are presented for a simulation with 27648 ions. Results for 8192 ions (squares) are also shown. Finally the dotted lines show the simple global fit of Eq. 8.

0.646 0.516 0.443 0.526 0.408 0.337 0.544 0.433 0.352

We see that Di /D0 increases for 12 C for small Γ in a way that is not described by Eq. 8. Indeed Fig. 4 shows that Di depends more strongly on Zi than Eq. 8 for small Γ. Finally we also perform MD simulations for a one component system of pure 16 O in order to compare to our multicomponent results. Results for MD simulations with 8192 and 27648 ions are shown in Table II

5

1.4 N=27648 N=8192 Fit

N=8192 27648 Fit

1.2 1 D/D0

Di/D0

1.5

1

0.8 0.6

0.5 0

0.4 0

1

0.5 −1 Γ

FIG. 4: (Color on line) Diffusion constants Di over D0 , see Eq. 7, versus the inverse of the coulomb parameter 1/Γ. Results for 12 C (solid black line), 16 O (dashed red), and 22 Ne (dotdashed blue) are presented for a simulation with 27648 ions. Results for 8192 ions (squares) are also shown. Finally the dotted lines show the simple global fit of Eq. 8. This disagrees with 12 C and 16 O data for small Γ

50

100

Γ

150

200

250

FIG. 5: (Color on line) Diffusion constant Di over D0 , see Eq. 7, versus the coulomb parameter Γ for pure oxygen. Results for a simulation with 27648 ions are shown as circles connected by a solid black line while the red squares show 8192 ion results. Finally the dashed line shows the simple global fit of Eq. 8 with hZi/Zi = 1.

IV.

CONCLUSIONS

16

TABLE II: Diffusion constant D16 O /D0 , for a pure O system from simulations with N = 8192 and 27648 ions in a (possibly metastable) fluid state. The diffusion constant is scaled with D0 = 3¯ ωp a2 /Γ4/3 . This is a simple fit to the original one component plasma results, see Eq. 7 in the text. Γ 2.470 6.175 12.35 24.70 49.40 82.33 123.50 187.12 247.00

D/D0 D/D0 N = 8192 N=27648 0.680 0.831 1.035 1.237 1.307 1.194 1.000 .439

0.828 1.045 1.274 1.330 1.223 1.014 0.703 0.441

and Fig. 5. There is good agreement between these two simulations and reasonable agreement with Eq. 8 with hZi/Zi = 1. Previous MD simulations for one component Yukawa fluids fit D with [15, 17, 19] 173 1.154 D = 0.0028 + 0.00525 − 1 . ωp a2 Γ

(9)

The ratio of Eq. 9 to Eq. 7 is very similar in form to Fig. 5, see Fig. 5 of Ref. [17]. However, we find a somewhat larger amplitude for the deviations of D from D0 .

Sedimentation of the neutron rich isotope 22 Ne may be an important source of gravitational energy during the cooling of white dwarf stars. This depends on the diffusion constant for 22 Ne in strongly coupled plasma mixtures. We have calculated self-diffusion constants Di from molecular dynamics simulations of carbon, oxygen, and neon mixtures. We find that, for strong coupling (coulomb parameter Γ > few), Di has a modest dependence on the charge −2/3 Zi of the ion species, Di ∝ Zi . However Di depends more strongly on Zi for weak coupling (smaller Γ). Our results for both a carbon, oxygen, neon mixture, and for a one component plasma can be fit by, 1 + 0.22Γ hZi 2/3 Di 0.62 ≈ 1.59 exp(−0.135Γ ) , ω ¯ p a2 Zi Γ4/3 (10) for Γ > few, see Eqs 8,7. Here ω ¯ p is the average plasma frequency and a is the ion sphere radius. We conclude that the self-diffusion constant DN e for 22 Ne in carbon, oxygen plasma mixtures is accurately known so that uncertainties in DN e should be unimportant for simulations of white dwarf cooling. All of our diffusion results have been for the liquid phase. We are not aware of any results for Di in solids. Often Di is arbitrarily assumed to be zero for a solid. In future work we will simulate Di in the solid phase, both for single component and multicomponent systems. We thank E. Brown, A. Cumming, and Z. Medin for helpful discussions. This research was supported in part

6 by DOE grant DE-FG02-87ER40365 and by the National Science Foundation through TeraGrid resources provided by National Institute for Computational Sciences, and

Texas Advanced Computing Center under grant TGAST100014.

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